Gravitational lensing of gravitational waves: universal characteristics of strongly lensed memory waveforms

TL;DR

This work analyzes the strong gravitational lensing of gravitational-wave memory, showing that lensing acts as a high-pass filter that renders memory signals oscillatory. The authors derive time-domain lensed memory expressions for all image types (I/III via a two-part structure and II via a single integral), unveil universal symmetry properties that are independent of lens models or binary parameters, and demonstrate that memory can be well approximated by a two-parameter step function. They propose a fast, two-step identification strategy to classify image type (favoring Type II) before employing full oscillatory templates for parameter estimation, with mismatches indicating substantial fidelity for many LISA-relevant events. The results suggest practical pathways to leveraging lensed memory for astrophysical and cosmological probes, including dark matter and cosmological constant measurements, using space-borne detectors.

Abstract

In this work, the strong lensing effect of the memory signal was considered. In the geometric optics limit, the lensed memory signal becomes oscillatory, while the unlensed is basically monotonic. This is because only the high frequency Fourier modes contribute to the lensed signal. Therefore, the strong lensing system serves as a high-pass filter for the memory signal. Due to the step function like behavior of the unlensed memory waveform, the lensed waveform possesses characteristic morphology that is dependent on the type of the image, and independent of the lens model and the binary system. That is, for each type of the lensed image, the lensed memory waveform has an approxmiate reflection symmetry about a symmetrical axis in the time domain. More specifically, for the type I and type III images, the lensed memory signals are nearly odd under the reflection, while the type II signal is roughly even. In addition, at the symmetrical axis, the sign of the slope for type I image is different from that for the type III image. These universal characteristic features would help determine the type of the lensed image. This is particularly because the memory waveform can be well approximated by a suitable step function, which involves two parameters. In addition, the detection of multiple images by the space-borne interferometer would also be helpful. Once the type of the lensed image is determined with the approximated memory waveform, one can use the appropriate waveform template for the oscillatory component of the gravitational wave to perform the parameter estimation.

Figures (6)

• Figure 1: Memory waveforms. Upper panel: the time-domain memory waveform $h_{\text{D}+}$ for the plus polarization at different $S_{1z}$'s. Lower panel: the frequency-domain memory waveform $\tilde{h}_{\text{D}}$ corresponding to the time-domain waveforms. The sensitivity curves of DECIGO, LISA, CE and ET are also displayed.
• Figure 2: The typical geometry of a gravitational lensing system. O is the observer, L is the lens, and S, S' are two nearby sources. The vertical lines represent the observer, the lens, and the source planes, respectively. $D_l$, $D_s$, and $D_{ls}$ are the distances between these planes. The horizontal dashed line, connecting O and L, is the optical axis. $\vec{\eta}$ is the position of the source S relative to the intersection of the optical axis and the source plane. $\vec{\xi}$ is the position of the deflecting point of the red trajectory relative to the lens L on the lens plane, while $\vec{\xi}'$ is for the magenta trajectory. $\beta$ is the misalignment angle for S.
• Figure 3: The lensed memory waveforms in the time domain. Upper panel: The lensed memory waveforms $h_\text{D}^{(1)}$ and $h_{\text{D}}^{(2)}$ from the two images at $x_1$ and $x_2$, respectively. If the first image were of type III, one would get the lensed waveform in the inset. Lower panel: The unlensed memory waveform $h_\text{D}$ and the interfered memory waveform $h_\text{D}^{(\text{L})}$. In both panels, the vertical line is at $T=\Delta T\approx2.8$ s.
• Figure 4: The expressions in the squared brackets of Eqs. \ref{['eq-lhh-td-mm']} and \ref{['eq-lhh-td-s']}. The red and blue curves correspond to the squared brackets in Eqs. \ref{['eq-lhh-td-mm']} and \ref{['eq-lhh-td-s']}, respectively.
• Figure 5: The actual memory waveform $\tilde{h}_\text{D}$ and the approximate one $\tilde{h}_H$. The black curve is LISA's noise curve.